# Simplified equations for FM (frequency modulation) synthesis and PM (phase modulation)?

Clicking on the links I find on the web it seems many authors don't express them neatly enough for me or they're even confusing the two. Can someone please clear this up?

Is frequency modulation

Ugh!

• Why 'carrier='? The carrier should be without modulation! – N74 Aug 1 '16 at 20:36
• @N74 Fixed and done. I think you can edit questions if you have enough points. – user23148 Aug 1 '16 at 20:39
• $x(t) = A(t) e^{i \phi(t)}$ : $A(t)$ is the time-varying amplitude, $\phi(t)$ the phase, $\omega(t) = \phi'(t)$ the time-varying (angular) frequency. for $x(t)$ to be a pitched sound, $A(t)$ has to be band-limited, and $\frac{\omega'(t)}{\omega(t)}$ too – user1952009 Aug 1 '16 at 20:47
• @user1952009 Am I 100% correct in assuming $phase modulation=sin(frequency*time-(phase+modulator))$. Can I check that off the list? – user23148 Aug 1 '16 at 20:56
• to be clear, AM is when the signal is in $A(t)$ and $w(t)$ is constant, and $FM$ is when the signal is in $\omega'(t) / \omega(t)$, and $A(t)$ is constant. phase modulation is almost the same as FM but high-pass filtered, with the signal in $\phi(t)$. AM is the most obvious way for transmitting a signal, while FM is the most robust for the radio when there are some interferances – user1952009 Aug 1 '16 at 22:20

If $t$ is time, $s(t)$ is the (appropriately scaled) signal, $\omega_0$ is the angular frequency, and $\phi_0$ is a phase offset, then phase modulation is $$t \mapsto \sin(\omega_0 t + \phi_0 + s(t))$$ Different signs for $s(t)$ and/or $\phi_0$ may be used, depending on conventions and context.
Frequency modulation is more complex to write down, because we want $s(t)$ to vary the derivative of the argument to the sine as a function of time. We get something like: $$t \mapsto \sin\bigl(\phi_0 + {\textstyle\int_0^t(\omega_0+s(u))du} \bigr)$$ which is the same as $$t \mapsto \sin\bigl(\omega_0t + \phi_0 + {\textstyle\int_0^t s(u)du} \bigr)$$
Your proposed $$t \mapsto \sin\bigl( (\omega_0+s(t))t + \phi_0 \bigr)$$ won't work -- it would look like phase modulation, with a modulation strength that increases monotonically and without bound depending on how long it is since $t=0$.